Theorem imaeq2 5103

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 5039	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 4992	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4768	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4768	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2292	1 |- ( A = B -> ( C " A ) = ( C " B ) )

Syntax hints:   -> wi 4   = wceq 1398   ran crn 4756   |` cres 4757   "cima 4758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-sn 3701  df-pr 3702  df-op 3704  df-br 4116  df-opab 4178  df-xp 4761  df-cnv 4763  df-dm 4765  df-rn 4766  df-res 4767  df-ima 4768

This theorem is referenced by:  imaeq2i  5105  imaeq2d  5107  fimadmfo  5605  ssimaex  5744  ssimaexg  5745  isoselem  6000  f1opw2  6270  supp0cosupp0fn  6481  fopwdom  7103  ssenen  7119  fiintim  7205  fidcenumlemrk  7238  fidcenumlemr  7239  sbthlem2  7242  isbth  7251  ennnfonelemp1  13246  ennnfonelemnn0  13262  ctinfomlemom  13267  ctinfom  13268  tgcn  15204  iscnp4  15214  cnpnei  15215  cnima  15216  cnconst2  15229  cnrest2  15232  cnptoprest  15235  txcnp  15267  txcnmpt  15269  metcnp3  15507